Beyond Beer's Law: Spectral Mixing Rules

Based on Beer's law, it is assumed that the absorbance of a mixture is that of the neat materials weighted by their relative amounts (linear mixing rule). In this contribution, we show that this is an assumption that holds only under various approximations for which no change of the chemical interactions is just one among several. To understand these approximations, which lead incrementally to different well known mixing rules, we finally derive the linear mixing rule from the Lorentz-Lorenz relation, with the first approximation that the local electric field is correctly described in this relation. Further levels of approximation are that the local field equals the applied field (Newton-Laplace mixing rule) and that the change of the index of refraction and, equivalently, absorption is weak (Gladstone-Dale/Arago-Biot mixing rule). Even then the linear mixing rule is only strictly valid if the indices of refraction in the transparency region at higher frequency than the absorption have the same value and the mixing is homogeneous relative to the resolving power of the light ("micro-homogeneous"). Under these preconditions, linear mixing of the individual absorbances is established. We illustrate the spectral differences between the different mixing rules, all of which are based on volume and not on mass fractions, with examples. For micro-heterogeneous samples, a different linear mixing rule governs the optical properties, which refers to the experimental quantities, reflectance, and transmittance. As a result, for such samples, mixtures of already comparably high content give only weak signals due to band flattening, which are hard to distinguish from baseline effects.